The filling problem in the cube

TL;DR

This paper establishes an isoperimetric inequality for filling cellular cycles in high-dimensional cubes, identifying optimal exponents and providing explicit examples of cycles that achieve these bounds.

Contribution

It proves a new isoperimetric inequality for cellular cycles in high-dimensional cubes and demonstrates the optimality of the exponent with explicit cycle examples.

Findings

Proved an isoperimetric inequality for cellular cycles in high-dimensional cubes.

Identified a family of cycles where the inequality's exponent is optimal.

Provided explicit examples demonstrating the sharpness of the inequality.

Abstract

We prove an isoperimetric inequality for filling cellular cycles in a high dimensional cube with cellular chains. In addition, we provide a family of cubical cellular cycles for which the exponent in the inequality is optimal.